Proposition 6.3.6.3. Let $f: X \rightarrow S$ be a morphism of simplicial sets. For every morphism $T \rightarrow S$, let $W_{T}$ denote the collection of all edges $w = (w_ T,w_ X)$ of the fiber product $T \times _{S} X$ for which $w_ T$ is a degenerate edge of $T$. The following conditions are equivalent:
- $(1)$
For every morphism of simplicial sets $T \rightarrow S$, the projection map $T \times _{S} X \rightarrow T$ exhibits $T$ as a localization of $T \times _{S} X$ with respect to $W_{T}$.
- $(2)$
The morphism $f$ is universally localizing, in the sense of Definition 6.3.6.1.
- $(3)$
For every simplex $\sigma : \Delta ^ n \rightarrow S$, the projection map $\Delta ^ n \times _{S} X \rightarrow \Delta ^ n$ exhibits $\Delta ^ n$ as a localization of $\Delta ^{n} \times _{S} X$ with respect to some collection of edges of $\Delta ^ n \times _{S} X$.
- $(4)$
For every simplex $\sigma : \Delta ^ n \rightarrow S$, the projection map $\Delta ^ n \times _{S} X \rightarrow \Delta ^ n$ exhibits $\Delta ^ n$ as a localization of $\Delta ^{n} \times _{S} X$ with respect to $W_{\Delta ^ n}$.
Proof.
The implications $(1) \Rightarrow (2) \Rightarrow (3)$ are immediate. We next show that $(3)$ implies $(4)$. Let $\sigma $ be an $n$-simplex of $S$, and suppose that the projection map $\pi : \Delta ^ n \times _{S} X \rightarrow \Delta ^ n$ exhibits $\Delta ^ n$ as a localization of $\Delta ^{n} \times _{S} X$ with respect to some collection of edges $W$. Since $\Delta ^ n$ is an $\infty $-category in which every isomorphism is an identity morphism, the diagram $\pi $ must carry each edge of $W$ to a degenerate edge of $\Delta ^ n$: that is, we have $W \subseteq W_{\Delta ^ n}$. Applying Corollary 6.3.1.22, we deduce that $\pi $ also exhibits $\Delta ^ n$ as a localization of $\Delta ^ n \times _{S} X$ with respect to $W_{\Delta ^{n}}$.
We now complete the proof by showing that $(4)$ implies $(1)$. Let us say that a simplicial set $T$ is good if, for every morphism $T \rightarrow S$, the projection map $T \times _{S} X \rightarrow T$ exhibits $T$ as a localization of $T \times _{S} X$ with respect to $W_{T}$. Assume that condition $(4)$ is satisfied, so that every standard simplex $\Delta ^{m}$ is good. We wish to show that every simplicial set $T$ is good. Using Proposition 6.3.4.1, we see that the collection of good simplicial sets is closed under filtered colimits; we may therefore assume without loss of generality that $T$ is finite. If $T = \emptyset $, the result is obvious. We may therefore assume that $T$ has dimension $n$ for some integer $n \geq 0$. We proceed by induction on $n$ and on the number of nondegenerate $n$-simplices of $T$. Fix a nondegenerate $n$-simplex $\sigma : \Delta ^ n \rightarrow T$. Using Proposition 1.1.4.12, we see that there is a pushout square of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^ n \ar [r] \ar [d] & \Delta ^ n \ar [d]^{\sigma } \\ T' \ar [r] & T, } \]
where $T'$ is a simplicial set of dimension $\leq n$ having fewer nondegenerate $n$-simplices than $T$. By virtue of Proposition 6.3.4.2, to show that $T$ is good, it will suffice to show that the simplicial sets $\Delta ^ n$, $\operatorname{\partial \Delta }^ n$, and $T'$ are good. In the first case this follows from assumption $(4)$, and in the remaining cases it follows from our inductive hypothesis.
$\square$